Efficient c-Oriented Range Searching with DOP-Trees

Abstract

A c-dop is a c-oriented convex polytope, that is, a convex polytope whose edges have orientations that come from a fixed set of c orientations. In this paper we study dop-trees—bounding-volume hierarchies that use c-dops as bounding volumes—in the plane. We prove that for any set S of n disjoint c-dops in the plane, one can construct a dop-tree such that a range query with a c-dop as query range can be answered in O(n1/2 + ε + k) time, where k is the number of reported answers. This is optimal up to the factor O(nε). If the c-dops in S may intersect, the query time becomes O(n\(^{\rm 1-1/{\it c}}\)+k), which is optimal.

Keywords

Query Range Point Query Query Time Representative Point Splitting Line

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